Theorem ismndo 33671

Description: The predicate "is a monoid". (Contributed by FL, 2-Nov-2009.) (Revised by Mario Carneiro, 22-Dec-2013.) (New usage is discouraged.)

Hypothesis

Ref	Expression
ismndo.1	|- X = dom dom G

Assertion

Ref	Expression
ismndo	|- ( G e. A -> ( G e. MndOp <-> ( G e. SemiGrp /\ E. x e. X A. y e. X ( ( x G y ) = y /\ ( y G x ) = y ) ) ) )

Distinct variable groups:   x, G, y   x, X, y

Allowed substitution hints:   A( x, y)

Proof of Theorem ismndo

Step	Hyp	Ref	Expression
1		df-mndo 33666	. . 3 |- MndOp = ( SemiGrp i^i ExId )
2	1	eleq2i 2693	. 2 |- ( G e. MndOp <-> G e. ( SemiGrp i^i ExId ) )
3		elin 3796	. . 3 |- ( G e. ( SemiGrp i^i ExId ) <-> ( G e. SemiGrp /\ G e. ExId ) )
4		ismndo.1	. . . . 5 |- X = dom dom G
5	4	isexid 33646	. . . 4 |- ( G e. A -> ( G e. ExId <-> E. x e. X A. y e. X ( ( x G y ) = y /\ ( y G x ) = y ) ) )
6	5	anbi2d 740	. . 3 |- ( G e. A -> ( ( G e. SemiGrp /\ G e. ExId ) <-> ( G e. SemiGrp /\ E. x e. X A. y e. X ( ( x G y ) = y /\ ( y G x ) = y ) ) ) )
7	3, 6	syl5bb 272	. 2 |- ( G e. A -> ( G e. ( SemiGrp i^i ExId ) <-> ( G e. SemiGrp /\ E. x e. X A. y e. X ( ( x G y ) = y /\ ( y G x ) = y ) ) ) )
8	2, 7	syl5bb 272	1 |- ( G e. A -> ( G e. MndOp <-> ( G e. SemiGrp /\ E. x e. X A. y e. X ( ( x G y ) = y /\ ( y G x ) = y ) ) ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   A.wral 2912   E.wrex 2913   i^i cin 3573   dom cdm 5114  (class class class)co 6650   ExId cexid 33643   SemiGrpcsem 33659  MndOpcmndo 33665

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-dm 5124  df-iota 5851  df-fv 5896  df-ov 6653  df-exid 33644  df-mndo 33666

This theorem is referenced by:  ismndo1  33672